A magnetic origin for high temperature superconductivity in iron pnictides
Meng Wang, Chenglin Zhang, Xingye Lu, Guotai Tan, Huiqian Luo, Yu Song, Miaoyin Wang, Xiaotian Zhang, E. A. Goremychkin, T. G. Perring, T. A. Maier, Zhiping Yin, Kristjan Haule, Gabriel Kotliar, Pengcheng Dai
AA magnetic origin for high temperature superconductivity in iron pnictides
Meng Wang ∗ , Chenglin Zhang ∗ , Xingye Lu ∗ ,
1, 2
Guotai Tan, Huiqian Luo, Yu Song, Miaoyin Wang, Xiaotian Zhang, E.A. Goremychkin, T. G. Perring, T. A. Maier, Zhiping Yin, Kristjan Haule, Gabriel Kotliar, and Pengcheng Dai
2, 1 Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996-1200, USA ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, UK Center for Nanophase Materials Sciences and Computer Science and Mathematics Division,Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6494, USA Department of Physics, Rutgers University, Piscataway, NJ 08854, USA
In conventional Bardeen-Cooper-Schrie ff er (BCS) superconductors [1], superconductivity occurs when electrons formcoherent Cooper pairs below the superconducting transition temperature T c . Although the kinetic energy of paired elec-trons increases in the superconducting state relative to the normal state, the reduction in the ion lattice energy is su ffi cientto give the superconducting condensation energy ( E c = − N (0) ∆ / and ∆ ≈ (cid:126) ω D e − / N (0) V , where N (0) is the electron den-sity of states at zero temperature, (cid:126) ω D is the Debye energy, and V is the strength electron-lattice coupling) [2, 3, 13]. Foriron pnictide superconductors derived from electron or hole doping of their antiferromagnetic (AF) parent compounds[5–10], the microscopic origin for superconductivity is unclear [11]. Here we use neutron scattering to show that high- T c superconductivity only occurs for iron pnictides with low-energy ( ≤ meV or ∼ . k B T c ) itinerant electron-spin exci-tation coupling and high energy ( > meV) spin excitations. Since our absolute spin susceptibility measurements foroptimally hole-doped iron pnictide reveal that the change in magnetic exchange energy below and above T c can accountfor the superconducting condensation energy, we conclude that the presence of both high-energy spin excitations givingrise to a large magnetic exchange coupling J and low-energy spin excitations coupled to the itinerant electrons is essentialfor high- T c superconductivity in iron pnictides. For BCS superconductors, the superconducting condensation energy E c and T c are controlled by the strength of the Debyeenergy (cid:126) ω D and electron-lattice coupling V [1]. A material with large (cid:126) ω D and lattice exchange coupling is a necessary butnot a su ffi cient condition to have high- T c superconductivity. On the other hand, a soft metal with small (cid:126) ω D (such as lead andmercury) will also not exhibit superconductivity with high- T c . For unconventional superconductors such as iron pnictides, thesuperconducting phase is derived from hole and electron doping from their AF parent compounds [5–10]. Although the staticlong-range AF order is gradually suppressed when electrons or holes are doped into the iron pnictide parent compound suchas BaFe As [6–10], short-range spin excitations remain throughout the superconducting phase and are coupled directly withthe occurrence of superconductivity [4–6, 12, 13, 15–19]. For spin excitations mediated superconductors, the superconductingcondensation energy should be accounted for by the change in magnetic exchange energy between the normal ( N ) and super-conducting ( S ) phases at zero temperature. Within the t - J model [22], ∆ E ex ( T ) = J [ (cid:104) S i + x · S i (cid:105) N − (cid:104) S i + x · S i (cid:105) S ], where (cid:104) S i + x · S i (cid:105) is the dynamic spin susceptibility in absolute units at temperature T [3, 13].To determine how high- T c superconductivity in iron pnictides is associated with spin excitations, we consider the phasediagram of electron and hole-doped iron pnictide BaFe As (Fig. 1a) [10]. In the undoped state, BaFe As forms a metallic low-temperature orthorhombic phase with collinear AF structure as shown in the inset of Fig. 1a. Inelastic neutron scattering (INS)measurements have mapped out spin waves throughout the Brillouin zone, and determined the e ff ective magnetic exchange cou-plings [2]. Upon doping electrons to BaFe As by partially replacing Fe with Ni to induce superconductivity in BaFe − x Ni x As with maximum T c ≈
20 K at x e = . <
80 meV) spin waves in the parent compounds are broadened andform a neutron spin resonance coupled to superconductivity [16–19], while high-energy spin excitations remain unchanged [6].With further electron-doping to x e ≥ .
25, superconductivity is suppressed and the system becomes a paramagnetic metal (Fig.1a) [24]. For hole-doped Ba − x K x Fe As [25], superconductivity with maximum T c = . x h ≈ .
33 [6] and pureKFe As at x h = T c = . . K . Fe As ( T c = . As ( T c = . Ni . As (Fig. 1d). If spin excita-tions are responsible for mediating electron pairing and superconductivity, the change in magnetic exchange energy between thenormal ( N ) and superconducting ( S ) state should be large enough to account for the superconducting condensation energy [13].In previous INS work on powder [12, 13] and single crystals [4] of hole-doped Ba − x K x Fe As , low-energy spin excitationswere found to be dominated by a resonance coupled to superconductivity. In fact, density functional theory (DFT) calculationsbased on sign reversed quasiparticle excitations between hole Fermi surface pocket near the Brillouin zone center and electronpocket near the zone corner (see supplementary information) [11] predict correctly the longitudinally elongated spin excitationsfrom Q AF = (1 ,
0) for optimally hole-doped iron pnictide (inset in Fig. 1a) [4, 19]. For pure KFe As ( x h = a r X i v : . [ c ond - m a t . s up r- c on ] M a r ( <
14 meV) spin excitations become longitudinally incommensurate from Q AF (inset in Fig. 1a) [15]. These results, as well aswork on electron-doped iron pnictides that reveal transversely elongated spin excitations from Q AF [5, 6, 19], have shown thatthe low energy nematic-like spin dispersion in iron-based superconductors can be accounted for by itinerant electrons on holeand electron nested Fermi surfaces [10].Here we use INS to show that the e ff ective magnetic exchange couplings in hole-doped Ba . K . Fe As only soften slightlyfrom that of AF BaFe As [2] and electron-doped BaFe . Ni . As superconductor (Fig. 1f) [6]. The e ff ect of hole-dopingin BaFe As is to suppress high-energy spin excitations and transfer the spectral weight to low-energies that couple to theappearance of superconductivity (Fig. 1h). This is qualitatively consistent with theoretical methods based on DFT and dynamicmean filed theory (DMFT, see supplementary information) [7]. By using the INS measured magnetic exchange couplings andspin susceptibility in absolute units, we calculate the superconductivity-induced lowering of magnetic exchange energy and findit to be about seven times larger than the superconducting condensation energy determined from specific heat measurementsfor Ba . K . Fe As [15]. These results are consistent with spin excitations mediated electron pairing mechanism [13]. Forthe nonsuperconducting electron overdoped BaFe . Ni . As , we find that while the e ff ective magnetic exchange couplings onlyreduce slightly compared with that of optimally electron-doped BaFe . Ni . As (Fig. 1g) [6], the low-energy spin excitations( <
50 meV) associated with hole and electron pocket Fermi surface nesting disappear, thus revealing the importance of Fermisurface nesting and itinerant electron-spin excitation coupling to the occurrence of superconductivity (Fig. 1h). Finally, forheavily hole-doped KFe As with low- T c superconductivity (Fig. 1b), there are only incommensurate spin excitations below ∼
25 meV and the correlated high-energy spin excitations prevalent in electron-doped and optimally hole-doped iron pnictidesare completely suppressed (Fig. 1e), indicating a dramatic softening of e ff ective magnetic exchange coupling (inset Fig. 1h).Therefore, high- T c superconductivity in iron pnictides requires two fundamental ingredients: a large e ff ective magnetic exchangecoupling [13], much like large Debye energy for high- T c BCS superconductors, and a strong itinerant electrons-spin excitationscoupling from Fermi surface nesting [11], like electron-phonon coupling in BCS superconductors. The presence of correlatedelectronic states exhibiting both local and itinerant properties is essential for the mechanism of superconductivity [10].To substantiate the key conclusions of Fig. 1, we present the two-dimensional (2D) constant-energy images of spin excitationsin the ( H , K ) plane at di ff erent energies for KFe As (Figs. 2a-2c), Ba . K . Fe As (Figs. 2d-2f), and BaFe . Ni . As (Figs.2g-2i) above T c . In previous INS work on KFe As , longitudinal incommensurate spin excitations were found by triple axisspectrometer measurements for energies from 3 to 14 meV in the normal state [15]. While we confirmed the earlier work usingtime-of-flight INS for energies below E = ± E =
20 meV and disappear for energiesabove 25 meV (Fig. 2c). Therefore, there are no measurable correlated spin excitations for energies above 30 meV, indicatingthat the e ff ective magnetic coupling in KFe As has reduced about 90% compared with that of BaFe As (see supplementaryinformation). For Ba . K . Fe As , spin excitations at E = ± Q AF as expected fromDFT calculations (Fig. 2d) [4, 19]. At the resonance energy ( E = ± T c (Fig.2e). On increasing energy further to E = ±
10 meV, spin excitations change to transversely elongated from Q AF similar tospin excitations in optimally electron-doped superconductor BaFe . Ni . As (Fig. 2f) [6]. Figures 2g-2i summarize similar 2Dconstant-energy images of spin excitations for nonsuperconducting BaFe . Ni . As. At E = ± ±
10 meV(Fig. 2h), there are no correlated spin excitations near the Q AF . Upon increasing energy to E = ±
10 meV (Fig. 2i), we seeclear spin excitations transversely elongated from Q AF (Fig. 2i).Figures 3a-3d show 2D images of spin excitations in BaFe . Ni . As at E = ±
10, 112 ±
10, 157 ±
10, and 214 ±
10 meV,respectively. Figures 3e-3h show wave vector dependence of spin excitations at energies E = ± , ± , ±
10, and195 ±
10 meV, respectively, for Ba . K . Fe As . Similar to spin waves in BaFe As [2], spin excitations in BaFe . Ni . As and Ba . K . Fe As split along the K -direction for energies above 80 meV and form rings around Q = ( ± , ±
1) positions nearthe zone boundary, albeit at slightly di ff erent energies. Comparing spin excitations in Figs. 3a-3d for BaFe . Ni . As with thosein Figs. 3e-3h for Ba . K . Fe As in absolute units, we see that spin excitations in BaFe . Ni . As extend to slightly higherenergies and have larger intensity above 100 meV.To understand the wave vector dependence of spin excitations in hole-doped Ba . K . Fe As , we have carried out therandom phase approximation (RPA) calculation of the dynamic susceptibility in a pure itinerant electron picture using methoddescribed before [5]. Figures 3i and 3j show RPA calculations of spin excitations at E =
70 and 155 meV, respectively, forBa . K . Fe As assuming that hole doping induces a rigid band shift [5]. The outcome is in clear disagreement with Figs.3e and 3g, indicating that a pure RPA type itinerant model cannot describe the wave vector dependence of spin excitations inhole-doped iron pinctides at high energies. For comparison with the RPA calculation, we also used a combined DFT and DMFTapproach [6, 7] to calculate the imaginary part of the dynamic susceptibility χ (cid:48)(cid:48) ( Q , ω ) in the paramagnetic state. Figures 3k and3i show calculated spin excitations at E =
70 and 155 meV, respectively. Although the model still does not agree in detail withthe data in Figs. 3e and 3g, it captures the trend of spectral weight transfer away from Q AF = (1 ,
0) on increasing the energy andforming a pocket centered at Q = (1 , , K ] and [ H ,
0] directions (see supple-mentary information), we establish the spin excitation dispersion along the two high symmetry directions for Ba . K . Fe As and compare with the dispersion of BaFe As (Fig. 1f) [2]. In contrast to the dispersion of electron-doped BaFe . Ni . As [6],we find clear softening of the zone boundary spin excitations in hole-doped Ba . K . Fe As from spin waves in BaFe As [2].We estimate that the e ff ective magnetic exchange coupling in Ba . K . Fe As is reduced by about 10% from that of BaFe As (see supplementary information). Similarly, Figure 1g shows the dispersion curve of BaFe . Ni . As along the [1 , K ] directionplotted together with that of BaFe As [2]. For energies below ∼
50 meV, spin excitations are completely gapped marked in thedashed area probably due to the missing hole-electron Fermi pocket quasiparticle excitations [11, 28]. Based on the 2D spinexcitation images similar to Figs. 2a-2c, we plot in Fig. 1e the dispersion of incommensurate spin excitations in KFe As . Theincommensurability of spin excitations is weakly energy dependent below E =
12 meV but becomes smaller with increasingenergy above 12 meV (see supplementary information). Correlated spin excitations for energies above 25 meV are completelysuppressed as shown in the shaded area in Fig. 1e.To quantitatively determine the e ff ect of electron and hole doping on the overall spin excitations spectra, we calculate the localdynamic susceptibility per formula unit (f.u.) in absolute units, defined as χ (cid:48)(cid:48) ( ω ) = (cid:82) χ (cid:48)(cid:48) ( q , ω ) d q / (cid:82) d q [6], where χ (cid:48)(cid:48) ( q , ω ) = (1 / tr ( χ (cid:48)(cid:48) αβ ( q , ω )), at di ff erent energies for Ba . K . Fe As , BaFe . Ni . As , and KFe As . Figure 1h shows the outcometogether with previous data on optimally electron-doped superconductor BaFe . Ni . As [6]. While electron-doping up toBaFe . Ni . As does not change the spectral weight of high-energy spin excitations from that of BaFe . Ni . As , hole-dopingdramatically suppresses the high-energy spin excitations and shift the spectral weight to lower energies (Fig. 1h). For heavilyhole-doped KFe As , spin excitations are confined to energies below about E =
25 meV (inset in Fig. 1h). The reductionof the high energy spin spectral weight and its transfer to low energy with hole doping, but not with electron doping, is notnaturally explained by the band theory, and requires models which incorporate both the itinerant quasiparticles and the localmoment physics (see supplementary information) [7]. The hole doping makes electronic state more correlated, as local momentformation is strongest in the half-filled d shell, and mass enhancement larger thereby reducing the electronic energy scale in theproblem.Finally, to determine how low-energy spin excitations are coupled to superconductivity in Ba . K . Fe As , we carried out adetailed temperature dependent study of spin excitation at E = ± c -axis modulated low-energy( E < c -axis modulations.Figures 4a-4d show the 2D mapping of the resonance at T = , , , and 45 K, respectively. While the resonance revealsa clear oval shape at temperatures below T c (Figs. 4a and 4b), it changes into an isotropic circular shape abruptly at T c (Figs.4c and 4d) as shown by the dashed lines representing full-width-at-half-maximum (FWHM) of the excitations. Temperaturedependence of the resonance width along the [ H ,
0] and [1 , K ] directions in Fig. 4e reveals that the isotropic to anisotropictransition in momentum space occurs at T c . Figure 4f shows temperature dependence of the resonance from 9 K to 40 K,which vanishes at T c . Figure 4g plots temperature dependence of the mode energy together with the sum of the superconductinggaps from hole and electron pockets [28]. Figure 4h compares temperature dependence of the superconducting condensationenergy [15] with superconductivity induced intensity gain of the resonance. By calculating spin excitations induced changes inmagnetic exchange energy (see supplementary information) [13], we find that the di ff erence of magnetic exchange interactionenergy between the superconducting and normal state is approximately seven times larger than the superconducting condensationenergy [15], thus identifying AF spin excitations as the major driving force for superconductivity in Ba . K . Fe As .One way to quantitatively estimate the impact of hole / electron doping and superconductivity to spin waves of BaFe As is todetermine the energy dependence of the local moment and total fluctuating moments (cid:68) m (cid:69) [6]. From Fig. 1h, we see that hole-doping suppresses high-energy spin waves of BaFe As and pushes the spectral weight to resonance at lower energies. The totalfluctuating moment of Ba . K . Fe As below 300 meV is (cid:68) m (cid:69) = . ± . . ± . µ B per Fefor BaFe As and BaFe . Ni . As [6]. For comparison, BaFe . Ni . As and KFe As have (cid:68) m (cid:69) = . ± .
11 and 0 . ± . µ B per Fe, respectively. Therefore, the total magnetic spectral weights for di ff erent iron pnictides have no direct correlation withtheir superconducting T c ’s. From Fig. 1h, we also see that the spectral weight of the resonance and low-energy ( <
100 meV)magnetic scattering in Ba . K . Fe As is much larger than that of electron-doped BaFe . Ni . As . This is consistent witha large superconducting condensation energy in Ba . K . Fe As since its e ff ective magnetic exchange coupling J is only ∼
10% smaller than that of BaFe . Ni . As (Fig. 1h) [15, 29]. For electron-overdoped nonsuperconducting BaFe . Ni . As , thelack of superconductivity is due to the absence of low-energy spin excitations coupled to the hole and electron Fermi surfacenesting even though the e ff ective magnetic exchange remains large [11, 28]. This means that by eliminating [ (cid:104) S i + x · S i (cid:105) N −(cid:104) S i + x · S i (cid:105) S ], there is no magnetic driven superconducting condensation energy and thus no superconductivity. On the other hand,although the complete suppression of correlated high-energy spin excitations in KFe As can dramatically reduce the e ff ectivemagnetic exchange coupling in KFe As (Fig. 1e), one can still have superconductivity with reduced T c . If spin excitations area common thread of the electron pairing interactions for unconventional superconductors [13], our results reveal that both thelarge e ff ective magnetic exchange couplings and itinerant electron-spin excitation interactions are essential ingredients to achievehigh- T c superconductivity, much like large Debye energy and strength of electron-lattice coupling are necessary for high- T c BCS superconductors. Therefore, the presence of correlated electronic states exhibiting wave-particle duality is important forthe mechanism of high- T c superconductivity in iron pnictides. [1] Bardeen, J., Cooper, L. N., & Schrie ff er, J. R., Theory of Superconductivity. Phys. Rev. , 1175-1204 (1957).[2] Chester, G. V., Di ff erence between normal and superconducting states of a metal. Phys. Rev. , 1693-1699 (1956).[3] Scalapino, D. J. & White, S. R., Superconducting condensation energy and an antiferromagnetic exchange-based pairing mechanism.Phys. Rev. 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As super-conductor. Phys. Rev. Lett. , 107006 (2009).[18] Inosov, D. S., et al. , Normal-state spin dynamics and temperature-dependent spin-resonance energy in optimally doped BaFe . Co . As .Nat. Phys. , 178-181 (2010).[19] Park, J. T. et al., Symmetry of spin excitation spectra in tetragonal paramagnetic and superconducting phases of 122-ferropnictides. Phys.Rev. B , 134503 (2010).[20] Liu, M. S. et al. , Nature of magnetic excitations in superconducting BaFe . Ni . As . Nature Phys. , 376-381 (2012).[21] Luo, H. Q., et al. , Electron doping evolution of the anisotropic spin excitations in BaFe − x Ni x As . Phys. Rev. B , 024508 (2012).[22] Spalek, J., t - J model then and now: A personal perspective from the pioneering times. Acta Physica Polonica A , 409-424 (2007).[23] Harriger, L. W. et al. , Nematic spin fluid in the tetragonal phase of BaFe As . Phys. Rev. B , 054544 (2011).[24] Li, L. J. et al. , Superconductivity induced by Ni doping in BaFe As single crystals. New J. Phys. , 025008 (2009).[25] Avci, S. et al. , Magnetoelastic coupling in the phase diagram of Ba − x K x Fe As as seen via neutron di ff raction. Phys. Rev. B , 172503(2011).[26] Park, H., Haule, K., & Kotliar, G., Magnetic excitation spectra in BaFe As : a two-particle approach within a combination of the densityfunctional theory and the dynamical mean-field theory method. Phys. Rev. Lett. , 137007 (2011).[27] Popovich, P. et al. , Specific heat measurements of Ba . K . Fe As single crystals: evidence of a multiband strong-coupling supercon-ducting state. Phys. Rev. Lett. 105, 027003 (2010).[28] Richard, P., Sato, T., Nakayama, K., Takahashi, T., & Ding, H., Fe-based superconductors: an angle-resolved photoemission spectroscopyperspective. Rep. Prog. Phys. , 124512 (2011).[29] Zeng, B. et al. Specific heat of optimally doped Ba(Fe − x T M x ) As ( T M = Co and Ni) single crystals at low temperatures: A multibandfitting. Phys. Rev. B , 224514 (2012). Acknowledgements
Work at IOP is supported by the MOST of China 973 programs (2012CB821400, 2011CBA00110) andNSFC-11004233. The single crystal growth and neutron scattering work at UTK is supported by the U.S. DOE BES underGrant No. DE-FG02-05ER46202. The LDA + DMFT computations were made possible by an Oak Ridge leadership computingfacility director discretion allocation to Rutgers. The work at Rutgers is supported by DOE BES DE-FG02-99ER45761 (GK)and nsf-dmr 0746395 (KH). TAM acknowledges the Center for Nanophase Materials Sciences, which is sponsored at ORNL bythe Scientific User Facilities Division, BES, U.S. DOE.
Author contributions
This paper contains data from three di ff erent neutron scattering experiments in the group of P. D. lead byM.W. (Ba . K . Fe As ), C.L.Z (KFe As ), and X.Y.L (BaFe . Ni . As ). These authors made equal contributions to theresults reported in the paper. For Ba . K . Fe As , M.W., H.Q.L., E.A.G., and P.D. carried out neutron scatteringexperiments. Data analysis was done by M.W. with help from H.Q.L., and E.A.G.. The samples were grown by C.L.Z., M.W.,Y.S., X.Y.L., and co-aligned by M.W. and H.Q.L. RPA calculation is carried out by T.A.M. The DFT and DMFT calculationswere done by Z.P.Y., K.H., and G.K. Superconducting condensation energy was estimated by X.T.Z. For KFe As , the sampleswere grown by C.L.Z and G.T.T. Neutron scattering experiments were carried out by C.L.Z., T.G.P., and P.D. ForBaFe . Ni . As , the samples were grown by X.Y.L., H.Q.L., and coaligned by. X.Y.L and M.Y.W. Neutron scatteringexperiments were carried out by X.Y.L, T.G.P., and P.D. The data are analyzed by X.Y.L. The paper was written by P.D., M.W.,X.Y.L., C.L.Z. with input from T.A.M, K.H., and G.K.. All coauthors provided comments on the paper. Additional information
The authors declare no competing financial interests. Correspondence and requests for materialsshould be addressed to P.D., [email protected]
FIG. 1:
Summary of transport, magnetic, and neutron scattering results . Our experiments were carried out on the MERLIN andMAPS time-of-flight chopper spectrometers at the Rutherford-Appleton Laboratory, UK [2, 6]. We co-aligned 19 g of single crystals ofBa . K . Fe As (with in-plane and out-of-plane mosaic of ∼ ◦ ), 3 g of KFe As (with in-plane and out-of-plane mosaic of ∼ ◦ ), and 40g of BaFe . Ni . As (with in-plane and out-of-plane mosaic of ∼ ◦ ). Various incident beam energies were used as specified, and mostly with E i parallel to the c -axis. To facilitate easy comparison with spin waves in BaFe As [2], we defined the wave vector Q at ( q x , q y , q z ) as( H , K , L ) = ( q x a / π, q y b / π, q z c / π ) reciprocal lattice units (rlu) using the orthorhombic unit cell, where a = b = .
57 Å, and c = . . K . Fe As , a = b = .
43 Å, and c = . As , and a = b = . c = .
96 Å for BaFe . Ni . As . Thedata are normalized to absolute units using a vanadium standard with 20% errors [6]. (a) The electronic phase diagram of electron andhole-doped BaFe As [10]. The right inset shows crystal and AF spin structures of BaFe As with marked the nearest ( J a , J b ) and nextnearest neighbor ( J ) magnetic exchange couplings. The left insets show the evolution of low-energy spin excitations in Ba − x K x Fe As . (b,c)Temperature dependence of magnetic susceptibility for our KFe As and Ba . K . Fe As . (d) Temperature dependence of the resistivity forBaFe . Ni . As . (e,f,g) The filled circles are spin excitation dispersions of KFe As at 5 K, Ba . K . Fe As at 9 K, and BaFe . Ni . As at 5 K, respectively. The shaded areas indicate vanishing spin excitations and the solid lines show spin wave dispersions of BaFe As [2].(h) Energy dependence of χ (cid:48)(cid:48) ( ω ) for BaFe . Ni . As (dashed line), BaFe . Ni . As (green solid circles), Ba . K . Fe As below (Solid redcircles and solid red line) and above (open purple circles and solid lines) T c . The inset shows Energy dependence of χ (cid:48)(cid:48) ( ω ) for KFe As . Thevertical error bars indicate the statistical errors of one standard deviation. The horizontal error bars in (h) indicate the energy integration range. FIG. 2:
Constant-energy slices through magnetic excitations of KFe As , Ba . K . Fe As , and BaFe . Ni . As at di ff erent energies .The color bars represent the vanadium normalized absolute spin excitation intensity in the units of mbarn / sr / meV / f.u. Two dimensional imagesof spin excitations at 5 K for KFe As (a) E = ± E i =
20 meV. The right side incommensurate peak is obscured bybackground scattering. (b) 13 ± E i =
35 meV, (c) 53 ±
10 meV with E i =
80 meV. For Ba . K . Fe As at T =
45 K, images ofspin excitations at (d) E = ± E i =
20 meV, (e) 15 ± E i =
35 meV, and (f) 50 ± E i = ff erent energies. Images of spin excitations for BaFe . Ni . As at T = E = ± E i =
80 meV,(h) 30 ±
10 meV with E i =
450 meV, and (i) 59 ±
10 meV with E i =
250 meV. The white crosses indicate the position of Q AF . FIG. 3:
Constant-energy images of spin excitations in the 2D [ H , K ] plane at di ff erent energies for BaFe . Ni . As and Ba . K . Fe As and its comparison with RPA / DMFT calculations for Ba . K . Fe As . Spin excitations of BaFe . Ni . As at energy transfers of (a) E = ±
10 meV obtained with E i =
250 meV;(b) 112 ±
10 meV E i =
250 meV;(c) 157 ±
10 meV; (d) 214 ±
10 meV, all obtained with E i =
450 meV at 5 K. A flat backgrounds have been subtracted from the images. Spin excitations of Ba . K . Fe As at energy transfers of(e) E = ±
10 meV obtained with E i =
170 meV;(f) 115 ±
10 meV;(g) 155 ±
10 meV; (h) 195 ±
10 meV, all obtained with E i =
450 meV at 9K. Wave vector dependent backgrounds have been subtracted from the images. RPA calculations [5] of spin excitations for Ba . K . Fe As at (i) E =
70 meV and (j) E =
155 meV. DMFT calculations [6, 7] for Ba . K . Fe As at (k) E =
70 meV and (l) E =
155 meV.
FIG. 4:
Wave vector, energy, and temperature dependence of the resonance across T c = . K . Constant-energy ( E = ± T =
25, (b) 38, (c) 40, and (d)45 K obtained with E i =
35 meV. In order to make fair comparison of thescattering line shape at di ff erent temperatures, the peak intensity at each temperature is normalized to 1. The pink and green arrows in (a)mark wave vector cut directions across the resonance. The integration ranges are − . ≤ K ≤ . H ,
0] direction and 0 . ≤ H ≤ . , K ] direction. The FWHM of spin excitations are marked as dashed lines. (e) The FWHM of the resonance along the [ H ,
0] and[1 , K ] directions as a function of temperature across T c . (f) Energy dependence of the resonance obtained by subtracting the low-temperaturedata from the 45 K data, and correcting for the Bose population factor. (g) The black diamonds show temperature dependence of the sum ofhole and electron pocket electronic gaps obtained from Angle Resolved Photoemission experiments for Ba . K . Fe As [28]. The red solidcircles show temperature dependence of the resonance. (h) Temperature dependence of the superconducting condensation energy from heatcapacity measurements [15] and the intensity of the resonance integrated from 14 to 16 meV. The error bars indicate the statistical errors ofone standard deviation. Supplementary Information: A magnetic origin for high-temperature superconductivity in iron pnictides
Meng Wang ∗ , Chenglin Zhang ∗ , Xingye Lu ∗ , Guotai Tan, Huiqian Luo, Yu Song, Miaoyin Wang, Xiaotian Zhang, E. A. Gore-mychkin, T. G. Perring, T. A. Maier, Zhiping Yin, Kristjan Haule, Gabriel Kotliar, Pengcheng Dai ADDITIONAL DATA AND ANALYSIS
We first discuss detailed experimental results on electron-doped iron pnictides, focusing on comparison of electron over-doped nonsuperconducting BaFe . Ni . As with optimally electron-doped superconductor BaFe . Ni . As and antiferromag-netic (AF) BaFe As . SFigure 5 shows the evolution of Fermi surfaces as a function of increasing Ni-doping obtained from thetight-binding model of Graser et al. [1] and our measured low-energy spin excitations spectra for BaFe As , BaFe . Ni . As ,and BaFe . Ni . As . The absence of hole Fermi pocket near the zone center for BaFe . Ni . As means quasiparticle excitationsbetween the hole and electron pockets are not possible, thus eliminating low-energy spin excitations. SFigure 6 shows detailedcomparison of spin excitations at di ff erent energies for BaFe . Ni . As and BaFe . Ni . As . SFigure 7 plots the evolution ofspin excitations for BaFe . Ni . As in several Brillouin zones. SFigure 8 shows the identical wave vector versus energy cutsfor BaFe As [2] and BaFe . Ni . As . The spin excitations in BaFe . Ni . As are clearly absent below 50 meV, which is muchbigger than the 15 meV single ion spin anisotropy gap in BaFe As [3]. SFigure 9 shows the comparison of cuts along two high-symmetry directions for BaFe − x Ni x As at x e = , . , .
3. SFigure 10 plots the constant- Q cuts and spin correlation lengths forthese three samples. It is clear that the zone boundary spin excitations for electron-doped materials remain almost unchanged atleast up to x e = . SFig 5:
Schematics of Fermi surface evolution as a function of Ni-doping for BaFe − x Ni x As and corresponding spin excitations . (a,b,c)Evolution of Fermi surfaces with Ni-dopings of x e = , . , .
3. The d xz , d yz , and d xy orbitals for di ff erent Fermi surfaces are colored as red,green and blue, respectively. (d,e,f) Evolution of low-energy spin excitations for BaFe − x Ni x As with x e = , . , .
3. For data in (d) and (f), E i =
450 meV; (e) E i =
80 meV all with c -axis along incident beam direction. In Figure 2 of the main text, we have shown that the widths of spin excitations change with increasing energy above T c for Ba . K . Fe As . The resonance width also changes across T c (Figure 4 of the main text). To further illustrate the lineshape change between the normal and superconducting states, we need to determine the width ratio of spin excitations alongthe two high symmetry directions as a function of increasing energy above and below T c . We plot the fitting of full width athalf maximum (FWHM) of spin excitations along H and K directions in the normal state in SFig. 11 (a). It is a hour-glass likedispersion along [ H ,
0] direction, but a linear dispersion along [1 , K ] direction. On cooling below T c , the dispersion changesto a shape of flower vase in SFig. 11 (b). we plot in SFig. 11 (c) the energy dependence of the ratio ( K − H ) / ( K + H ) FWHM from 5 to 36 meV. In the normal state at T =
45 K, spin excitations at energies below the resonance have an oval shape withan elongated H direction, thus giving negative anisotropy ratio for energies below about 15 meV. On moving to the resonance1 SFig 6:
Comparison of spin excitations in absolute units for BaFe − x Ni x As with x e = . , .
3. Evolution of low-energy spin excitationsfor BaFe − x Ni x As with x e = . , .
3. (a) Data was obtained with E i =
30 meV for x e = . E i =
80 meV for x e = .
3. (b) E i =
80 meVfor x e = . E i =
250 meV for x e = .
3. (c) E i =
80 meV for x e = . E i =
450 meV for x e = .
3. (d) E i =
80 meV for x e = . E i =
250 meV for x e = .
3. (e,f,g,h) The corresponding cuts along the [1 , K ] direction for these two samples. energy at E =
15 meV, the scattering become isotropic and the isotropic scattering persist up to 30 meV. For energies above 30meV, transverse elongated scattering take over, giving positive anisotropy ratio. In our previous work [4], we reported that spinexcitations integrated from 10 to 18 meV display an longitudinally elongated oval shape in the normal state. This is consistentwith present work, which has much better energy resolution and better statistics for measured spin excitations. On cooling tothe superconducting state, scattering are essentially isotropic below the resonance, and change to the maximum anisotropy at theresonance energy. For energies above the resonance, the scattering change back to transverse elongated spin excitations above35 meV.In the main text, the integrated dynamic (local) susceptibilities for hole-doped Ba . K . Fe As , electron-dopedBaFe . Ni . As , and BaFe . Ni . As are shown in Fig. 1h. To compare spin excitations of Ba . K . Fe As withBaFe − x Ni x As at x e = , .
1, we show in SFig. 12 two dimensional constant-energy images at di ff erent energies in recip-rocal space for these materials. The magnetic scattering intensities for the three compounds at the same energy are normalizedto absolute units in the same color scale. The measurements for Ba . K . Fe As and BaFe . Ni . As were carried out on theMERLIN time-of-flight (TOF) chopper spectrometer at the Rutherford-Appleton Laboratory (RAL), UK. The BaFe As mea-2 SFig 7:
Spin excitations in absolute units for BaFe . Ni . As . (a-h) Evolution of spin excitations for BaFe . Ni . As all the way to thezone boundary. Data taken on MAPS at 5 K with E i =
250 meV for (b,d) and E i =
450 meV for (a,c,e,f,g,h). surements were carried out at MAPS, RAL. The incident beams were set to be parallel to the c -axis of the sample in all threeexperiments. Although spin excitation energies are coupled to momentum transfers along the c -axis in this scattering geometry,we note that MERLIN and MAPS experiments have almost identical L for spin excitation energy if the incident beam energyis the same. Therefore, the raw data with the same E i for Ba . K . Fe As , BaFe . Ni . As , and BaFe As can be compareddirectly. For energy transfer of E = ± E i =
35 meV for Ba . K . Fe As , 30 meVfor BaFe . Ni . As , and 80 meV for BaFe As . To compare the data quantitatively, the scattering intensity in 12(a,e,i) havebeen corrected by the magnetic form factor. Spin excitations for Ba . K . Fe As have the strongest intensity at the resonanceenergy ( E =
15 meV) and are rotated 90 ◦ in reciprocal space from the line-shape for BaFe . Ni . As . Spin waves in BaFe As has a clear anisotropy spin gap below 15 meV. For spin excitations at E = ± E i =
80 meV. Here, spin excitations for Ba . K . Fe As are transversely elongated but broader in reciprocal space comparedto those of BaFe . Ni . As and BaFe As (SFig. 12b, 12f, 12i). The scattering intensity is larger than that of BaFe . Ni . As ,but smaller than spin waves of BaFe As . However, the integrated dynamic susceptibility of Ba . K . Fe As is similar withthat of BaFe As , but larger than that of BaFe . Ni . As . At energies above E >
100 meV, spin excitations of Ba . K . Fe As are clearly weaker in intensity than that of BaFe . Ni . As and BaFe As . The constant-energy images have been subtracted bya radial background for Ba . K . Fe As and a constant background for BaFe . Ni . As and BaFe As .To quantitatively determine the dispersions of spin excitations in Ba . K . Fe As , BaFe . Ni . As , and BaFe As , we cutthrough [ H ,
0] and [1 , K ] directions of the two dimensional scattering images in SFig. 12. SFig. 13a-13c and 13d-13e showconstant-energy cuts at energies of E = ±
1, 45 ±
5, 70 ± H ,
0] and [1 , K ] directions for Ba . K . Fe As ,BaFe . Ni . As , and BaFe As , respectively. SFig. 13a,13d cuts have been corrected the e ff ect of Bose population factor andmagnetic form factor, since they have di ff erent incident beam energies and T =
45 K data. The cuts along the [1 , K ] direction at3 SFig 8:
The dispersion of spin excitations for BaFe − x Ni x As along [1 , K ] direction . (a, c) The dispersion cuts of BaFe As ( x =
0) with E i =
250 meV and E i =
450 meV along [1 , K ] direction. The data is from MAPS. (b, d) Identical dispersion cuts of BaFe . Ni . As ( x e = . Spin excitations cuts in absolute units for BaFe − x Ni x As . (a-f) Evolution of spin excitations along the [1 , K ] direction forBaFe − x Ni x As at x e = , . , .
3. (g-i) Similar cuts along the [ H ,
0] direction. E = ±
10, 155 ±
10 and 195 ±
10 meV reveal the weaker susceptibility at high energies for Ba . K . Fe As . The dashedblue lines in SFig. 13g, 13h indicate that spin excitations of Ba . K . Fe As disperse more rapidly and have a softened bandtop.Spin excitations disappear at the zone boundary [1 ,
1] in Ba . K . Fe As , BaFe . Ni . As , and BaFe As . The band top isgoverned by the e ff ective magnetic exchange couplings J ( J a , J b and J ), as defined in BaFe As [2]. To estimate the changeof J for hole-doped Ba . K . Fe As , we calculate the energy cut at [1 ,
1] by exploring the Heisenberg Hamiltonian of parentcompound. It turns out that J a , J b and J have comparable e ff ect on the band top. Based on the dispersion of Ba . K . Fe As ,4 SFig 10:
Spin excitations cuts in absolute units for BaFe − x Ni x As . (a,b) Constant- Q cuts of spin excitations for BaFe − x Ni x As at x e = , . , .
3. (c) Coherence lengths of spin excitations for these three samples.SFig 11:
Dispersion and anisotropy ratio of the low energy spin excitations for Ba . K . Fe As above and below T c . (a)Normal and(b)superconducting states spin excitation dispersion in full-width at half maximum ( FHW M ) along the [ H ,
0] and [1 , K ] directions. (c) Thefilled red circles show the anisotropy ratio at T =
45 K, and the filled blue squares are the same ratio at 9 K. The vertical error bars indicatethe statistical errors of one standard deviation. The solid lines are a guide to the eye. the e ff ective magnetic exchange J is found to be about 10 % smaller for Ba . K . Fe As compared with that of BaFe As (SFig. 14). For comparison, if we assume the band top for KFe As is around E =
25 meV, the e ff ective magnetic exchangeshould be about 90% smaller for KFe As . Of course, we know this is not an accurate estimation since spin excitations inKFe As are incommensurate and have an inverse dispersion. In any case, given the zone boundary energy of E ≈
25 meV, thee ff ective magnetic exchange couplings in KFe As must be much smaller than that of BaFe As . SFigure 15 shows additionaldata for KFe As that clearly reveal a dramatic reduction in magnetic scattering above E =
20 meV.
RANDOM PHASE APPROXIMATION (RPA) AND DFT + DMFT CALCULATIONS
As discussed in the main text, RPA calculation of the wave vector dependence of spin excitations in hole-dopedBa . K . Fe As is in clear disagreement with experiments (Fig. 3 of the main text). Although a combined DFT and DMFTapproach [6, 7] still does not agree in detail with the data (Fig. 3), it captures the trend of spectral weight transfer away from Q AF = (1 ,
0) on increasing the energy and forming a pocket centered at Q = (1 , SFig 12:
A comparison of constant-energy images of spin excitations for Ba . K . Fe As , BaFe . Ni . As , and BaFe As as a functionof increasing energy at low-temperature . The color bars represent the vanadium normalized absolute spin excitation intensity in the units ofmbarn / sr / meV / f.u. (a) E = ±
1, (b) E = ±
5, (c) E = ±
10, and (d) E = ±
10 meV are for Ba . K . Fe As at 9 K. (e-h) and(i-l) are identical images for BaFe . Ni . As at 5 K and BaFe As at 7 K, respectively.
16 show calculated local susceptibility in absolute units based on a combined DFT and DMFT approach for Ba . K . Fe As ,BaFe As , and BaFe . Ni . As [6], respectively. This theoretical method predicts that electron doping to BaFe As does nota ff ect the spin susceptibility at high energy ( E >
150 meV), while spin excitations in the hole doped compound beyond 100meV are suppressed by shifting the spectral weight to lower energies. This is in qualitative agreement with our absolute intensitymeasurements (Fig. 1h). The reduction of the high energy spin spectral weight and its transfer to low energy with hole doping,but not with electron doping, is not naturally explained by the band theory, and requires models which incorporate both theitinerant quasiparticles and the local moment physics. The hole doping makes electronic state more correlated, as local momentformation is strongest in the half-filled d shell, and mass enhancement larger thereby reducing the electronic energy scale in theproblem.Our theoretical DFT + DMFT method for computing the magnetic excitation spectrum employs the abinitio full potentialimplementation of the method, as detailed in [8]. The DFT part is based on the density functional theory (DFT) code of Wien2k6
SFig 13:
Constant-energy cuts of spin excitations as a function of increasing energy . (a-c) are the cuts along the [ H ,
0] direction and(d-i) are along the [1 , K ] direction for Ba . K . Fe As at 9 K (red filled circles), 45 K (half filled purple squares), BaFe . Ni . As at 5 K(solid blue line) and BaFe As at 7 K (solid black line). (a) E = ± E i =
35 meV for Ba . K . Fe As and E i =
30 forBaFe . Ni . As , (b) E = ± = E = ± E i =
250 meV are cuts along [ H ,
0] direction. (d-f) are thesame constant-energy cuts along [1 , K ] direction. (g) E = ±
10, (h) E = ±
10, (i) E = ±
10 meV are from E i =
450 meV data.The solid lines are Gaussian fits to the data. (a,d) have been corrected the Bose population factor and magnetic form factor.SFig 14:
The e ff ect of magnetic exchange couplings J ( J a , J b and J ) on the band top of spin excitations . The black line is energy cut at(0 . < H < .
2, 0 . < H < .
2) r.l.u for BaFe As in the Heisenberg spin wave model [2]. The red line is for Ba . K . Fe As with 10 % softenband top. The blue line is a similar estimation for KFe As assuming zone boundary is around E =
25 meV. [9]. The DMFT method requires solution of the generalized quantum impurity problem, which is here solved by the numericallyexact continuous-time quantum Monte Carlo method [10, 11]. The Coulomb interaction matrix for electrons on iron atom wasdetermined by the self-consistent GW method in Ref. [12], giving U = J = . + DMFT method. The dynamical magnetic susceptibility χ (cid:48)(cid:48) ( Q , E ) is computedfrom the ab initio perspective by solving the Bethe-Salpeter equation, which involves the fully interacting one particle Greensfunction computed by DFT + DMFT, and the two particle vertex, also computed within the same method (for details see Ref.[7]). We computed the two-particle irreducible vertex functions of the DMFT impurity model, which coincides with the local7
SFig 15:
The disappearance of sin excitations above 25 meV for KFe As . Using E i =
80 meV, we can still see clear incommensurate spinexcitations at (a) E = . ± . E = . ± . E >
25 meV. two-particle irreducible vertex within DFT + DMFT method. The latter is assumed to be local in the same basis in which theDMFT self-energy is local, here implemented by projection to the mu ffi n-tin sphere. SFig 16:
RPA and LDA + DMFT calculated local susceptibility for di ff erent iron pnictides . RPA and LDA + DMFT calculations of χ (cid:48)(cid:48) ( ω )in absolute units for Ba . K . Fe As comparing with earlier results for BaFe As and BaFe . Ni . As [6]. MAGNETIC EXCHANGE ENERGY AND SUPERCONDUCTING CONDENSATION ENERGY FOR BA . K . FE AS In a neutron scattering experiment, we measure scattering function S ( q , E = (cid:126) ω ) which is related to the imaginary part ofthe dynamic susceptibility via S ( q , ω ) = [1 + n ( ω, T )] χ (cid:48)(cid:48) ( (cid:126) q , ω ), where [1 + n ( ω, T )] is be Bose population factor. The magneticexchange coupling and the imaginary part of spin susceptibility are related via the formula [13]: (cid:104) (cid:126) S i · (cid:126) S j (cid:105) = π g µ B (cid:90) d (cid:126) q (2 π ) (cid:90) d ω [1 + n ( ω, T )] χ (cid:48)(cid:48) ( (cid:126) q , ω ) cos [ (cid:126) q · ( (cid:126) i − (cid:126) j )] , (1)8where g = g -factor. The magnetic exchange energy can be written as E ex = (cid:88) < i , j > J i j (cid:104) (cid:126) S i · (cid:126) S j (cid:105) = π g µ B (cid:90) d (cid:126) q (2 π ) (cid:90) d ω { (cid:88) i J a [1 + n ( ω, T )] χ (cid:48)(cid:48) ( (cid:126) q , ω ) cos ( q x ) + (cid:88) i J b [1 + n ( ω, T )] χ (cid:48)(cid:48) ( (cid:126) q , ω ) cos ( q y ) + (cid:88) i J [1 + n ( ω, T )] χ (cid:48)(cid:48) ( (cid:126) q , ω )[ cos ( q x + q y ) + cos ( q x − q y )] } . (2)Here we have assumed an anisotropic model for the e ff ective magnetic exchange coupling [2], di ff erent from the case of copperoxide superconductors [13]. J a is the e ff ective magnetic coupling strength between two nearest sites along the a direction,while J b is that along the b direction, and J is the coupling between the next nearest neighbor sites. Hence we are able toobtain the change in magnetic exchange energy between the superconducting and normal states by the experimental data of χ (cid:48)(cid:48) ( (cid:126) q , ω ) in both states. Strictly speaking, we want to estimate the zero temperature di ff erence of the magnetic exchange energybetween the normal and the superconducting states, and use the outcome to compare with the superconducting condensationenergy [13]. Unfortunately, we do not have direct information on the normal state χ (cid:48)(cid:48) ( (cid:126) q , ω ) at zero temperature. Nevertheless,since our neutron scattering measurements at low-energies showed that the χ (cid:48)(cid:48) ( (cid:126) q , ω ) are very similar below and above T c nearthe AF wave vector Q AF = (1 , ,
1) and only a very shallow spin gap at Q = (1 , ,
0) (see Figs. 1f and 1h in [4]), we assume thatthere are negligible changes in χ (cid:48)(cid:48) ( (cid:126) q , ω ) above and below T c at zero temperature for energies below 5 meV. For spin excitationenergies above 6 meV, Bose population factors between 7 K and 45 K are negligibly small. In previous work on optimally dopedYBa Cu O . superconductor, we have assumed that spin excitations in the normal state at zero temperature are negligibly smalland thus do not contribute to the exchange energy [14].The directly measured quantity is the scattering di ff erential cross section d σ d Ω dE k i k f = γ r e ) π g µ B | F ( (cid:126) Q ) | [1 + n ( ω, T )] χ (cid:48)(cid:48) ( (cid:126) q , ω ) , (3)where k i and k f are the magnitudes of initial and final neutron momentum and F ( (cid:126) Q ) is the Fe magnetic form factor, and( γ r e ) = . · sr − .The quantity γ r e ) π g µ B χ (cid:48)(cid:48) ( (cid:126) q , E ) in both superconducting and normal states can be fitted by a Gaussian A s ( n ) e − [ ( qx − σ x , s ( n ) + q y σ y , s ( n ) ] forresonance wave vector (1 ,
0) and by cutting the raw data. The outcome is summarized in the table : E σ x , s σ y , s A s σ x , n σ y , n A n .
050 0 .
060 6 .
017 0 .
115 0 .
076 6 . .
059 0 .
066 7 .
318 0 .
109 0 .
080 5 . .
077 0 .
077 9 .
789 0 .
154 0 .
100 5 . .
092 0 .
083 12 .
001 0 .
160 0 .
107 5 . .
121 0 .
097 14 .
674 0 .
145 0 .
124 5 . .
153 0 .
106 16 .
792 0 .
125 0 .
130 6 . .
167 0 .
118 12 .
141 0 .
152 0 .
157 5 . .
173 0 .
136 8 .
856 0 .
131 0 .
128 5 . .
165 0 .
155 6 .
802 0 .
134 0 .
146 5 . .
182 0 .
177 3 .
393 0 .
161 0 .
165 3 . where the unit of E is meV and that of A s ( n ) is mbarn · meV − · sr − · Fe − . For the case below 5 meV, we assume that A n decreasesto zero linearly with energy and A s = A n (see Fig. 1h in Ref. [4]), while the σ ’s keep the values at 5 meV. The assumption isshown in SFig. , where the resonance is seen at E =
15 meV.9
SFig 17: Assumed A s ( n ) below 5 meV. Because the condensation energy is only defined at zero temperature, we take T = (cid:104) (cid:126) S i · (cid:126) S i + x (cid:105) s − (cid:104) (cid:126) S i · (cid:126) S i + x (cid:105) n = − . , (cid:104) (cid:126) S i · (cid:126) S i + y (cid:105) s − (cid:104) (cid:126) S i · (cid:126) S i + y (cid:105) n = . , (cid:88) l = x ± y ( (cid:104) (cid:126) S i · (cid:126) S i + l (cid:105) s − (cid:104) (cid:126) S i · (cid:126) S i + l (cid:105) n ) = − . J a S = . , J b S = − . , J S = . , (5)which are 10% smaller than that of BaFe As [2] and we estimate S to be close to [16, 17]. Hence the exchange energy changeis ∆ E ex = − .
66 meV / Fe . (6)The condensation energy U c for optimally doped Ba . K . Fe As can be calculated to be U c = − . / mol = − . . × − . × f . u . = − . . × − × . × Fe = − .
09 meV / Fe (7)from the specific heat data of Ref. [15]. Therefore, we have the ratio ∆ E ex / U c ≈ .
4, meaning that the change in the magneticexchange energy is su ffi cient to account for the superconducting condensation energy in Ba . K . Fe As . We note that asimilar calculation for heavy Fermion superconductor CeCu Si also reveals that the change in magnetic exchange energy issu ffi cient to account for the superconducting condensation energy [18].We thank M. S. Liu and L. W. Harriger for providing the BaFe . Ni . As and BaFe As raw data, respectively.0 [1] Graser, S. et al ., Spin fluctuations and superconductivity in a three-dimensional tight-binding model for BaFe As . Phys. Rev. B ,214503 (2010).[2] Harriger, L. W. et al. , Nematic spin fluid in the tetragonal phase of BaFe As . Phys. Rev. B , 054544 (2011).[3] Matan, K, Morinaga, R., Iida, K., and Sato, T. J., Anisotropic itinerant magnetism and spin fluctuations in BaFe As : A neutron scatteringstudy. Phys. Rev. B , 054526 (2009).[4] Zhang, C. L. et al. , Neutron scattering studies of spin excitations in hole-doped Ba . K . Fe As superconductor. Scientific Reports ,115 (2011).[5] Luo, H. Q., et al. , Electron doping evolution of the anisotropic spin excitations in BaFe − x Ni x As . Phys. Rev. B , 024508 (2012).[6] Liu, M. S. et al. , Nature of magnetic excitations in superconducting BaFe . Ni . As . Nature Phys. , 376-381 (2012).[7] Park, H., Haule, K., & Kotliar, G., Magnetic excitation spectra in BaFe As : a two-particle approach within a combination of the densityfunctional theory and the dynamical mean-field theory method. Phys. Rev. Lett. , 137007 (2011).[8] Haule, K., Yee, C.-H., Kim, K., Dynamical mean-field theory within the full-potential methods: Electronic structure of CeIrIn , CeCoIn , and CeRhIn . Phys. Rev. B , 195107 (2010).[9] Blaha, P., Schwarz, K., Madsen, G. K. H., Kvasnicka, K., & Luitz, J., Wien2K, Karlheinz Schwarz, Technische Universitat Wien, Austria(2001).[10] Haule, K., Quantum Monte Carlo impurity solver for cluster dynamical mean-field theory and electronic structure calculations withadjustable cluster base. Phys. Rev. B , 155113 (2007).[11] Werner, P., Comanac, A., de’ Medici, L., Troyer, M., & Millis, A. J., Continuous-Time Solver for Quantum Impurity Models. Phys. Rev.Lett. , 076405 (2006).[12] Kutepov, A., Haule, K., Savrasov, S. Y., & Kotliar, G., Self-consistent GW determination of the interaction strength: Application to theiron arsenide superconductors. Phys. Rev. B , 045105 (2010).[13] Scalapino, D. J., A Common Thread: the pairing interaction for the unconventional superconductors. Rev. Mod. Phys. , 1383 (2012).[14] Woo, H. et al. , Magnetic energy change available to superconducting condensation in optimally doped YBa Cu O . . Nat. Phys. , 600(2006).[15] Popovich, P. et al. , Specific Heat Measurements of Ba . K . Fe As Single Crystals: Evidence for a Multiband Strong-Coupling Super-conducting state. Phys. Rev. Lett. , 027003 (2010).[16] Liu, M., et al. , Nature of magnetic excitations in superconducting BaFe . Ni . As . Nature Phys. , 376 (2012).[17] Zhao, J., et al. , Spin waves and magnetic exchange interactions in CaFe As . Nature Phys. , 555 (2009).[18] Stockert, O., et al. , Magnetically driven superconductivity in CeCu Si . Nature Phys.7